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Determinantal Ideals and Their Betti Numbers–A Survey

橋本光靖 (MITSUYASU HASHIMOTO) (Nagoya Univ.)

Abstract

This note is an introduction to the ring-theoretical approach to the study ofdeterminantal varieties, especially to the study of minimal free resolutions of deter-minantal ideals.

1 Determinantal Rings as ASL’s

Let $A$ be a noetherian ring, $I$ an ideal of $A$ , and $M$ a finitely generated A-module. Wedefine the I-depth of $M$ to be $\min\{i|Ext_{A}^{i}(A/I, M)\neq 0\}$ and denote it by depth(I, $M$ ).

If $A$ is a local ring with the maximal ideal $m$ , then depth $(\mathfrak{m}, M)$ is sometimes denotedby depth $M$ . In this case, we have depth $M\leq\dim M$ for $M\neq 0$ , where $\dim M$ is theKurll dimension of $A/ann_{A}M$ . We say that $M$ is Cohen-Macaulay when the equalityholds, or $M=0$. We say that the local ring $A$ is Cohen-Macaulay when so is $A$ as anA-module. A noetherian ring (which may not be local) $A$ is said to be Cohen-Macaulaywhen its localization at any maximal ideal is Cohen-Macaulay local.

Cohen-Macaulay property is one of the most important notion in the modern commu-tative ring theory.

Lemma 1.1 Let $A$ be a d-dimensional graded K-algebra ($K$ a field) generated by finitedegree one elements. Then, the following hold.

1 $A$ is Cohen-Macaulay if and only if depth$(A_{+}, A)=d$, where $A_{+}$ is the ideal of $A$

consisting of all degree positive elements.

2 ($K$ is assumed to be infinite) Let $\theta_{1},$

$\ldots$ , $\theta_{d}$ be degree one elements such that $A$ is afinite module over $K[\underline{\theta}]=K[\theta_{1}, \ldots, \theta_{d}]\subset A$ (such $\theta_{1},$

$\ldots,$$\theta_{d}$ do exist). Then, $A$ is

Cohen-Macaulay if and only if $A$ is $a$ free $K[\underline{\theta}]$ -module (hence, this condition doesnot depend on the choice of $\theta_{1},$

$\ldots,$$\theta_{d}$).

3 Let $a_{1},$ $\ldots,$$a_{f}$ be the degree one generator of $A$ as a K-algebra so that the map

$S=K[x_{1}, \ldots, x_{r}]arrow K[a_{1}, \ldots, a_{r}]=A(x_{i}-\rangle a_{i})$

is a surjective map of graded K-algebras. Then, $A$ is Cohen-Macaulay if and onlyif $pd_{S}A=r-d$ , where pd denotes the projective dimension.

数理解析研究所講究録第 857巻 1994年 40-50

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Gorenstein property is also important homological property. A noetherian local ring$A$ is said to be Gorenstein when its self-injective dimension is finite. A noetherian ringis said to be Gorenstein when its localization at any maximal ideal is Gorenstein. AnyGorenstein ring is Cohen-Macaulay, but the converse is not true in general.

Lemma 1.2 Let $A$ be a d-dimensional Cohen-Macaulay graded K-algebra ($K$ a field)generated by finite degree one elements. Then, the following hold.

1 $A$ is Gorenstein if and only if $Ext_{A}^{d}(A/A_{+}, A)\cong K$ .

2 Let $F_{A}(t)= \sum_{i>0}(\dim\tau^{F}A_{i})t^{i}$ . Then, $(1-t)^{d}F_{A}(t)$ is a polynomial in $t$ , say, $h_{0}+h_{1}t+$

. . . $+h_{s}t^{s}(h_{s}^{-}\neq 0)$ . If $A$ is Gorenstein, then $h_{s}=1$ . The converse is true when $A$

is an integral domain.

3 Let $a_{1},$ $\ldots,$$a_{r}$ be degree-one generators of $A$ , and consider $A$ as a module over $S=$

$K[x_{1}, \ldots, x_{r}]$ . Then, the following are equivalent.

a $A$ is Gorenstein.$bExt_{S}^{r-d}(A, S)$ is cyclic as an S-module.$b$ ‘ $Ext_{S}^{r-d}(A, S)\cong A$ as an S-module.

For a graded K-algebra $A$ , a graded A-module $M$ is said to be free when $M$ is a directsum of modules of the form $A(i)$ , where $A(i)$ is simply $A$ as an A-module, and the gradingis given by $A(i)_{j}=A_{i+j}$ . Clearly, a free module is projective in the category of gradedA-modules. Assume that $A$ is generated by finite elements of positive degree. For afinitely generated graded A-module $M$ and its subset $S=\{m_{1}, \ldots, m_{r}\},$ $S$ generates $M$

if and only if the image of $S$ generates $M/A_{+}M$ (an analogue of Nakayama’s lemma).So, $S$ is a set of minimal generators if and only if its image in $M/A_{+}M$ is a K-basis.

Let $R$ be a commutative ring with unity. For a matrix $(a_{i,j})\in Mat_{m,n}(R)$ with coef-ficients in $R$ and a positive integer $t$ , we define the determinantal ideal $I_{t}((a_{i,j}))$ of thematrix $(a_{i_{)}j})$ to be the ideal of $R$ generated by all t-minors of $(a_{i,j})$ .

We are $lnterested$ in the generic case here. Let $S=R[x_{ij}]_{1\leq i\leq m,1\leq j\leq n}$ be a polynomialring over $R$ in $mn$ variables. We set $X=(x_{ij})\in Mat_{m,n}(S)$ . The ideal $I_{t}=I_{t}(X)\subset S$

is considered to be a generic determinantal ideal. When we consider $S$ as a coordinatering of the affine space $Mat_{m,n}(R)$ , the ideal $I_{t}$ defines the closed subscheme $Y_{t}$ , the spaceof $m\cross n$ matrices whose rank is smaller than $t$ (because the rank of a matrix is smallerthan $t$ if and only if its all t-minors vanish). The following is a fundamental theorem ondeterminantal ideals.

Theorem 1.3 (Hochster-Eagon [HE]) Let $R$ be noetherian. The following hold.

1. $\dim S/I_{t}=\dim R+mn-(m-t+1)(n-t+1)$ .

2. The ideal $I_{t}$ is perfect (of codimension $(m-t+1)(n-t+1)$). Namely, we have

$depth_{S}(I_{t}, S)=pd_{S}S/I_{t}=(m-t+1)(n-t+1)$ .

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3. $S/I_{t}$ is R-flat.4. If $R$ is a domain, then so is $S/I_{t}$ .

5. If $R$ is normal, then so is $S/I_{t}$ .

Where pd denotes the projective dimension, and depth $(I_{t}, S)= \min\{i|Ext_{S}^{i}(S/I_{t}, S)\neq$

$0\}$ . There are some different proof Of this theorem. In this section, we give a (sketch of a)purely algebraic (or combinatorial) proof of the theorem which uses the theory of ASL’s.The lecture note [BV] gives a systematic account on this treatment.

The general theory tells us that it suffices to prove the following provided we haveproved that $S/I_{t}$ is R-flat.

Corollary 1.4 Assume that $R$ is a field. Then, we have $S/I_{t}$ is a Cohen-Macaulaynormal domain of dimension $mn-(m-t+1)(n-t+1)$ .

Definition 1.5 Let $R$ be a commutative ring, and $P$ a finite poset ( $=partially$ orderedset). We say that $A$ is a (graded) ASL (algebra with straightening lows) on $P$ over $R$ ifthe followings hold.

ASL-O An injective map $Prightarrow A$ is given, $A$ a graded R-algebra generated by $P$ , andeach element of $P$ is homogeneous of positive degree. We call a product of elementsof $P$ a monomial in $P$ . Formally, a monomial $M$ is a map $Parrow N_{0}$ , and we denote$M= \prod_{x\in P}x^{M(x)}$ so that it also stands for an element of $A$ . A monomial in $P$ ofthe form

$x_{i_{1}}\cdots x_{i_{l}}$

with $x_{i_{1}}\leq\cdots\leq x_{i_{l}}$ is called standard.

ASL-I The set of standard monomials in $P$ is an R-free basis of $A$ .

ASL-2 For $x,$ $y\in P$ such that $x\not\leq y$ and $y\not\leq x$ , there is an expression of the form

(1.6) $xy= \sum_{M}c_{M}^{xy}M$$(c_{M}^{xy}\in R)$

where the sum is taken over all standard monomials $M=x_{1}\cdots x_{r_{M}}(x_{1}\leq\cdots\leq$

$x_{r_{M}})$ with $x_{1}<x,$ $y$ and $\deg M=\deg(xy)$ .

The expression (1.6) in (ASL-2) condition is called the straightening relations of $A$ .The most simple example of an ASL on $P$ over $R$ is the Stanley-Reisner ring $R[P]=$$R[x|x\in P]/(xy|x\not\leq y, y\not\leq x)$ . The (ASL-2) condition is satisfied with letting theright-hand side zero. The Stanley-Reisner rings play central r\^ole in the theory of ASL.

Theorem 1.7 ([DEP]) Let $R$ be a commutative ring, $P$ a finite poset, and $A$ an $ASL$ on$P$ over R. Then, there is a sequence of $ASLs$ on $P$ over R $A=A_{0},$ $A_{1},$

$\ldots$ , $A_{m}=R[P]$

and an ideal $I_{i}$ of $A_{i}$ for each $i<m$ such that $A_{i+1}=G_{I_{i}}A_{i}$ for $i<m$ .

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Here, for a ring $A$ and its ideal $I,$ $G_{I}(A)$ denotes the associated graded ring $A[t^{-1}I,t]/(t)$ .Usually, the associated graded ring $G_{I}(A)$ is worse than $A$ . Hence, by the theorem, if$R[P]$ enjoys good property, then so does any ASL on $P$ over $R$ .

Corollary 1.8 If $R[P]$ is an integral domain (resp. Cohen-Macaulay, normal, Goren-stein), then any $ASL$ on $P$ over $R$ enjoys the same property.

As is clear, ASL’s on $P$ over a field have the same Hilbert function provided we give thesame degree to each element in $P$ . So it is completely determined only by the combinato-rial information on $P$ (because $H_{R}(n,$ $R[P])=\#\{standard$ monomials in $P$ of degree $n\}$ ).The corollary is not a good criterion of integrality, normality or Gorenstein property, be-cause $R[P]$ rarely satisfies these conditions. However, the corollary gives a good criterionof Cohen-Macaulay property.

Proposition 1.9 If $R$ is Cohen-Macaulay and if $P$ is a distributive lattice, then $R[P]$ isCohen-Macaulay.

For the proof of these results, see [DEP].As a result, the determinantal ring $S/I_{t}$ has a structure of an ASL on a distributive

lattice over $R$ , where $S=R[x_{ij}]_{1\leq i\leq m,1\leq J\leq n}$ . This shows that $S/I_{t}$ is R-flat (by ASL-I)and that $S/I_{t}$ is Cohen-Macaulay when so is $R$ .

First, we introduce an ASL structure into $S$ .We set

$\Omega_{s}=\{[i_{1}, \ldots, i_{s}; j_{1}, \ldots,j_{s}]|1\leq i_{1}<\cdots<i_{s}\leq m, 1\leq j_{1}<\cdots<j_{s}\leq n\}$

and $\Omega=\bigcup_{s=1}^{\min(m,n)}\Omega_{s}$ . We introduce an order structure into $\Omega$ . For elements $d=$

$[i_{1}, \ldots, i_{s};j_{1}, \ldots,j_{s}]$ and $d’=[i_{1}’, \ldots, i_{s’}’; j_{1}’, \ldots,j_{s}’]$ of $\Omega$ , we say that $d\leq d’$ if $s\geq s’$

and if $i_{l}\leq i_{l}’,$ $j_{l}\leq j_{l}’$ for $1\leq l\leq s’$ . It is easy to see that $\Omega$ is a distributive lattice withthis order structure.

We have a map $\Omegaarrow S$ given by

$[i_{1}, \ldots, i_{s};j_{1}, \ldots,j_{s}]rightarrow\det(x_{i_{a},j_{\beta}})_{1\leq\alpha,\beta\leq s}$ .

Lemma 1.10 With the structure above, $S$ is an $ASL$ on $\Omega$ over $R$ with the stmighteningrelation of the form

$ab=(a \wedge b)(a\vee b)+\sum_{c,d}u_{cd}^{ab}cd(+ve)$$(u_{cd}^{ab},v\in R)$

for each $a=[a_{1}, \ldots, a_{s};a_{1}’, \ldots, a_{s}’]$ and $b=[b_{1}, \ldots, b_{s’};b_{1}’, \ldots , b_{s}’]$ , where $a \wedge b=\inf(a, b)$ ,$a \vee b=\sup(a, b)$ , the sum is taken over all $c\in\Omega_{l}$ and $d\in\Omega_{l’}$ such that $c<a\wedge b$

and that $l+l’=s+s’$ . $ffa_{1},$$\ldots,$ $a_{s},$ $b_{1},$

$\ldots,$$b_{s’}$ are all distinct with its rearmngement

in the increasing order is $c_{I},$ $\ldots,$ $c_{s+s’}$ , and if $a_{1}’,$$\ldots$ , $a_{s}’,$ $b_{1}’,$

$\ldots$ , $b_{s}’$ , are all distinct withits rearmngement in the increasing order is $c_{1}’,$

$\ldots$ , $c_{s+s’}$ , then the term $ve(v\in R)$ mayappear in the right-hand side, where $e=[c_{1}, \ldots , c_{s+s’}; c_{1}’, \ldots, c_{s+s’}’]$ . The grading of $S$ isthe usual grading ($i.e$ , each $x_{ij}$ is degree one).

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This is proved using the Laplace expansion rule. See for example, [ABW2]. The ASLstructure above is good with the determinantal ideals $I_{t}$ .

For a poset $P$ and its subset $Q$ , we say that $Q$ is a poset ideal of $P$ when for any $x\in P$

and $y\in Q,$ $x\leq yi_{1}npliesx\in Q$ .

Lemma 1.11 Let $A$ be an $ASL$ on $P$ over $R$ with the stmightening relation $(1.\theta)$ . If$Q$ is a poset ideal of $P$ , then $A/I_{Q}$ is an $ASL$ on $P-Q$ over $R$ with the stmighteningrelation

$xy= \sum_{M}c_{M}^{xy}M$$(c_{M}^{xy}\in R)$

where $I_{Q}$ is the ideal $(x|x\in Q)$ in A generated by $Q$ and the sum is taken over all $M$

that appears in (1.6) such that no element in $Q$ appears in $M$ .

The proof is straightforward. Applying this lemma to the ASL $S$ on $\Omega$ and the posetideal $\Omega_{\geq\ell}=\bigcup_{s\geq t}\Omega_{s}$ of $\Omega$ , we conclude that $S/I_{t}$ is an ASL on $\Omega_{<t}=\Omega-\Omega_{\geq t}$ . Thus, $S/I_{t}$

is R-flat for any $R$ by (ASL-I). Moreover, it is easy to see that $\Omega_{<t}$ is a sublattice of $\Omega$ ,and hence is a distributive lattice. This shows that $S/I_{t}$ is Cohen-Macaulay when so is$R$.

It remains to show that $S/I_{t}$ is a normal domain when $R$ is a field. There is a goodcriterion of normality for ASL’s on distributive lattices due to Ito.

Theorem 1.12 ([Ito, Corollary]) Assume that $R$ is a Cohen-Macaulay normal do-main. Let $A$ be an $ASL$ over a distributive lattice $L$ with the stmightening relation

$xy=(x \wedge y)(x\vee y)+\sum_{M}c_{M}^{xy}M(c_{M}^{xy}\in R)$,

where the sum is taken over standard monomials $M=x_{1}\cdots x_{r_{M}}$ which have the samedegree as $xy$ with $x_{I}<x\wedge y$ . Then, $A$ is a Cohen-Macaulay normal domain.

The determinantal ring $S/I_{t}$ satisfies the assumption of this criterion, so it is a normaldomain. It is straightforward to see that rank $\Omega_{<t}=mn-(m-t+1)(n-t+1)-1$ sothat $\dim S/I_{t}=\dim R+mn-(m-t+1)(n-t+1)$ , and the proof of Theorem 1.3 iscompleted.

Hibi [Hib] defined the algebra $\mathcal{R}_{R}[L]=R[x\in L]/(xy-(x\wedge y)(x\vee y))$ for distributivelattices, and showed that this algebra is a Cohen-Macaulay normal domain. The algebra$\mathcal{R}_{R}[L]$ is called the Hibi ring of $L$ over $R$ . It follows that any distributive lattice isintegral. He posed a question that an ASL on $L$ with some good straightening relationis a normal domain [Hib, p.103]. Ito’s criterion is a good answer to this question.

For the Gorenstein property, Hibi completely determined when Hibi ring is Gorenstein.

Theorem 1.13 ([Hib, p.107]) Let $A$ be as in Theorem 1.12. Then, $A$ is Gorenstein ifand only if $R$ is Gorenstein, and $P$ is pure, where $P$ is the set ofjoin-irreducible elementsin L. That is,

$P=\{x\in L|\#\{y\in L|\#\{z\in L|y\leq z\leq x\}=2\}=1\}$ .

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Note that if the theorem is true for Hibi rings, then the theorem is true in general by2 of Lemma 1.2. It is obvious that $x=[a_{1}, \ldots, a_{s};b_{1}, \ldots , b_{s}]\in\Omega_{<t}$ is join-irreducible ifand only if one of the following is satisfied (we assume $t \leq\min(m,$ $n)$ ).

1. $x=[1, \ldots,s;1, \ldots, s](1\leq s\leq t-2)$

2. $x=[1, \ldots , i, m-s+i+1, \ldots, m;1, \ldots, s](1\leq s\leq t-1,0\leq i\leq s-1)$

3. $x=[1, \ldots, s;1, \ldots, i, n-s+i+1, \ldots, n](1\leq s\leq t-1,0\leq i\leq s-1)$

From this, it is not so difficult to show that $\Omega_{<t}$ is pure if and only if $t=1(\Omega<t=\emptyset)$

or $m=n$ .

Corollary 1.14 $S/I_{t}$ is Gorenstein if and only if $R$ is Gorenstein, and $t=1$ or $m=n$ .

2 A Mininal Free ResolutionThere has been much interest in determinantal ideals from the viewpoint of homologicalalgebra. Among them, the following is an interesting problem.

Problem 2.1 1. Construct a minimal free resolution of $S/I_{t}$ as a graded S-module.

2. Assume that $R$ is a field. Calculate the graded Betti numbers

$\beta_{ij}^{R}=\dim_{R}[Tor_{i}^{S}(S/S_{+}, S/I_{t})]_{j}$ ,

where $S_{+}=I_{1}=(x_{ij})$ , and $[$ $]_{j}$ denotes the degree $j$ component of a gradedS-module.

Here, a graded S-complex (i.e., a chain complex in the category of graded S-modules)

$F$ :. . . $arrow F_{i}arrow F_{i-1}\partial_{j}arrow\cdotsarrow F_{0}arrow 0$

is said to be a free resolution of a graded S-module $M$ when each $F_{i}$ is free, $H_{i}(F)=0$

$(i>0)$ and $H_{0}(F)=M$ . It is called minimal when the boundary maps of $S/S_{+}\otimes F$ areall zero. A graded minimal free resolution is unique up to isomorphism. It exists whenthe base ring $R$ is a field.

Since $S/I_{t}$ is free as an R-module, we have $Tor_{i}^{R}(\Lambda f, S/I_{t})=0$ for $i>0$ and any R-module $M$ . Hence, if $F$ is a projective resolution of $S/I_{t}$ over the base ring $R$ , and if $R’$ isan R-algebra, then $R’\otimes_{R}F$ is a projective resolution of $R’\otimes_{R}S/I_{t}$ . If $F$ is graded minimalfree, then so is $R’\otimes_{R}S/I_{t}$ . So, if 1 of the problem is solved for the ring of integers $Z$ ,then 1 is solved for any $R$ , because we can get the resolution by base change $R\otimes_{Z}?$ .

Let $F$ be a graded minimal free resolution of $S/I_{t}$ . Then, $H_{i}(S/s_{+}\otimes_{S}F)=S/S_{+}\otimes F_{i}$

is an R-free module, and we have

$\infty>rank_{R}Tor_{i}^{S}(S/S_{+}, S/I_{t})=rank_{S}F_{i}$.

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Note that the right hand side is invariant under the base change. In particular, for anyR-algebra $K$ which is a field, we have $\beta_{i}^{K}=rank_{S}F_{i}$ . Thus, the problem 2 is easier than1 (for example, if 1 is solved for any field, then 2 is completely solved).

Assume that $R$ is a field. Since $S/I_{t}$ is Cohen-Macaulay of dimension $\dim S-(m-t+$$1)(n-t+1)$ , we have $pd_{S}S/I_{t}=(m-t+1)(n-t+1)$ . We set $h=(m-t+1)(n-t+1)$ .Then, we have $\beta_{h}^{R}\neq 0$ and $\beta_{i}^{R}=0$ for $i>h$ . The ring $S/I_{t}$ is Gorenstein if and only if$\beta_{h}=1$ by Lemma 1.2. Let $F$ be a graded minimal free resolution of $S/I_{t}$ . Then, we have

$H_{i}(Hom_{S}(F, S))=Ext_{S}^{i}(S/I_{t}, S)=0$

unless $i=-h$ by Lemma 1.1, since $S/I_{t}$ is Cohen-Macaulay of codimension $h$ . So thecomplex $Hom_{S}(F, S)[-h]$ ( $[]$ denotes the shift of the degree as a chain complex) is aminimal free resolution of the S-module $Ext_{S}^{h}(S/I_{t}, S)$ . When $S/I_{t}$ is Gorenstein, wehave $Ext_{S}^{h}(S/I_{t}, S)\cong S/I_{t}$ . This shows that $Hom_{S}(F, S)[-h]$ is a graded minimal freeresolution of $S/I_{t}$ (the grading as a graded S-module may be different, so we should say$Hom_{S}(F, S)(a)[h]$ is a graded minimal free resolution of $S/I_{t}$ for some $a\in Z$ ). This showsthat

$F_{i}\cong Hom_{S}(F, S)[h]_{i}(a)=Hom_{S}(F_{h-i}, S)(a)$ ,

and we have $\beta_{i}=\beta_{h-i}$ .Why is the problem a problem? First, constructing a graded minimal free resolution of

$S/I$ as an S-module (for a homogeneous polynomial ring $S=K[x_{1}, \ldots, x_{r}]$ over a field$K$ and its homogeneous ideal $I$ ) has been considered as an ultimate aim of homologicalstudy of the algebra $S/I$–knowing a minimal free resolution yields ample informationon the ring in question. For example, $S/I$ is Cohen-Macaulay if and only if $\beta_{i}(S/I)=0$

for $i>\dim S-\dim S/I$ . It is Gorenstein if and only if it is Cohen-Macaulay and$\beta_{\dim S-\dim S/I}(S/I)=1$ . So the Betti numbers $\beta_{i}$ of an algebra contain alot of informationof the algebra (however, nowadays the progress of the theory of commutative algebraprovides us a lot of tools for studying important homological properties (such as Cohen-Macaulay property) of commutative algebras without constructing a resolution).

Secondly, the theory of the resolution of determinantal ideals is an interaction be-tween the theory of commutative algebra, combinatorics and the representation theoryof algebraic groups, and is interesting itself.

The number $\beta_{i}^{h’}$ depends only on the characteristic $p$ of $K$ , so we also write $\beta_{i}^{p}$ .When there exists a graded minimal free resolution $F$ of $S/I_{t}$ over the ring of integers

so that the resolution is obtained by base change for an arbitrary ring? Clearly, if sucha resolution exists over $Z$ , then $\beta_{i}^{p}$ is independent of $p$ . The converse is true.

Lemma 2.2 ([Rob, Chapter 4, Proposition 2], [HK, Proposition II.3.4]) Assumcthat $R$ is a noetherian reduced ring such that any finitely generated projective $R$ moduleis free. Let $A=R[x_{1}, \ldots, x_{n}]$ be a homogeneous polynomial ring over $R_{f}$ and $M$ afinitely generated gmded A-module which is flat as an R-module. Then, the following areequivalent for any $i\geq 0$ .

1 There exists a graded minimal free $co$ mplex

$0arrow F_{i+1}arrow F_{i}\partial_{i+1}arrow\partial$; . . . $arrow F_{1}arrow\ulcorner_{0}\partial_{1}arrow 0$

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such that $H_{0}F=M$ and $H_{k}F=0$ for $1\leq k\leq i$ .

2 For any $0\leq k\leq i$ and $j\in N_{0}$ , the numbers

$\beta_{kj}^{K}(M)^{dej}=\dim_{R/\mathfrak{M}}[Tor_{k}^{R/\mathfrak{M}\otimes_{R}A}(R/\mathfrak{M}\otimes_{R}A/A_{+}, R/\mathfrak{M}\otimes_{R}M)]_{j}$

is independent of the maximal ideal SEJt of $R$ , where $[]_{j}$ denotes the degree $j$ compo-nent of a gmded A-module.

3 For any $0\leq k\leq i$ , the Betti numbers $\beta_{k}^{K}(M)=\beta_{k}(R/\mathfrak{M}\otimes_{R}M)$ (over the field$K=R/M)$ is independent of the maximal ideal $\mathfrak{M}$ of $R$ .

4 For any $0\leq k\leq i,$ $Tor_{k}^{A}(A/A_{+}, M)$ is a free R-module.

Thus, there exists a graded minimal free resolution of $S/I_{t}$ over $Z$ if and only if $\beta_{i}^{\rho}(S/I_{t})$

is independent ofp for any i.

Problem 2.3 Are the Betti numbers $\beta_{i}^{p}(S/I_{t})$ independent of the characteristic?

Known approaches to the problem of the resolutions of determinantal ideals more or lessdepend on representation theory of GL. Let $V=R^{n}$ and $W=R^{m}$ . Then, the polynomialring $S=R[x_{ij}]_{1\leq i<m,1\leq j\leq n}$ is identified with the symmetric algebra $S(V\otimes W)$ , on which$G=GL(V)\cross GL\overline{(}W)$ acts. It is clear that $I_{t}$ is invariant under the action of $G$ .

Among various tools in the representation theory of GL, Schur modules and Schurcomplexes are very important.

Let $R$ be a commutative ring which contains the field of rationals $Q$ , and $C$ a finitefree R-complex (i.e., bounded R-complex with each term finite free). For $n>0$ , thesymmetric group $\mathfrak{S}_{n}$ acts on $C^{\cross n}$ by

$\sigma(a_{1}\otimes\cdots\otimes a_{n})=(-1)^{\Sigma_{1<j,\sigma i>\sigma g}\deg\langle a_{i})\deg(a_{j})}a_{\sigma^{-1}1}\otimes\cdots\otimes a_{\sigma}-1_{\mathcal{R}}$

for $\sigma\in \mathfrak{S}_{n}$ .For a partition (i.e., a weakly decreasing sequence of non-negative integers) A $=$

$(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{l})$ with $\Sigma_{i}\lambda_{i}=n$ , we set $L_{\lambda}C$ $:=Hom_{6_{n}}(s_{\overline{\lambda}}, C^{\otimes n})$ , where $s_{\overline{\lambda}}$ is the Spechtmodule (see e.g., [Gr]) of A (A $=(\tilde{\lambda}_{1},\tilde{\lambda}_{2}, \ldots)$ is the transpose of $\lambda$ , namely, the partitiongiven by $\tilde{\lambda}_{i}=\#\{j|\lambda_{j}\geq i\}$ ). This complex was used effectively, and the resolutionof determinantal ideals over the field of characteristic zero was constructed [Las], [Nls],[PW1].

It seems to be difficult to extend the definition of Schur complex $L_{\lambda}C$ (with goodproperty) of a finite free complex to the general base ring $R$ . However, there is a goodextension of the notion of Schur complex of a map (i.e., a complex of length at mostone) to the general base ring [ABW2]. For a map of finite free R-modules $\varphi$ : $Farrow E$ ,the Schur complex $L_{\lambda}\varphi$ is defined. The definition is compatible with the base extension.Namely, for any map of commutative rings $Rarrow R’$ , there is a canonical isomorphism ofR’-complexes

$R’\otimes_{R}L_{\lambda}\varphi\cong L_{\lambda}(R’\otimes_{R}\varphi)$ .

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Moreover, the definition of $L_{\lambda}\varphi$ agrees with that of Nielsen’s when the base ring $R$

contatins Q.The characteristic-free Schur complex is used to construct the minimal free resolution of

$S/I_{t^{r}}(r\geq 1)$ for $t= \min(m, n)$ . Using this, Akin, Buchsbaum and Weyman constructedthe minimal free resolution of $S/I_{t}$ for the case $t= \min(m, n)-1$ [ABW2].

Using characteristic-free representation theory developed by Akin, Buchsbaum andWeyman, Kurano [Kur] obtained the following result.

Theorem 2.4 The second Betti $number\beta_{2}^{1K}$ of the determinantal ring $S/I_{t}$ is independentof the base field $K$ .

In the proof of the theorem, the characteristic-free Cauchy’s formula [ABW2] playedthe central r\^ole. Cauchy’s formula for the characteristic zero case is stated as follows. Let$R\supset Q$ , and $V$ and $W$ be finite free R-modules. Then, for $r\geq 0$ , we have an isomorphismof $G=GL(V)\cross GL(W)$ -modules

$S_{r}(V \otimes W)\cong\bigoplus_{\lambda}L_{\lambda}V\otimes L_{\lambda}W$,

where the sum is taken over all partitions $\lambda=(\lambda_{1}, \lambda_{2}, \ldots)$ such that $\sum_{i}\lambda_{i}=r$ . Note thateach summand of the right-hand side is irreducible as a G-module or $0$ .

The characteristic-free version is stated using the characteristic-free Schur modules.After that, Kurano and the author extended the characteristic-free Cauchy’s formulato the chain complex version [HK], and proved that Problem 2.3 is true for the case$m=n=t+2$. After that, Problem 2.3 was solved negatively.

Theorem 2.5 ([Hasl]) We have $\beta_{3}^{z/3Z}>\beta_{3}^{Q}$ when $2 \leq t\leq\min(m, n)-3$ .

After that, the author proved that there exists a graded minimal free resolution of $S/I_{t}$

over $Z$ when $t= \min(m, n)-2$ [Has2]. Thus, we have

Theorem 2.6 There exists a graded minimal free resolution of $S/I_{t}$ if and only if $t=1$

or $t \geq\min(m, n)-2$ .

In the proof of the theorem, a certain class of subcomplexes of the Schur complex ofthe identity map $L_{\lambda}id_{F}$ , called the t-Schur complexes, was studied.

The t-Schur complexes are used to calculate the Betti numbers of other class of deter-minantal ideals [Has3], [Has4].

References[ABWI] K. Akin, D. A. Buchsbaum and J. Weyman, Resolutions of determinantal ideals:

the submaximal minors, Adv. in Math. 39 (1981), 1-30.

[ABW2] K. Akin, D. A. Buchsbaum and J. Weyman, Schur functors and Schur com-plexes, Adv. in Math. 44 (1982), 207-278.

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[Bbm] D. A. Buchsbaum, A New construction of the Eagon-Northcott complex, Adv.in Math. 34 (1979), 58-76.

[BV] W. Bruns and U. Vetter, “Determinantal rings,” Lect. Notes Math. 1327,Springer Verlag (1988).

[DEP] C. De Concini, D. Eisenbud and C. Procesi, “hodge algebras,” ast\’erisque 91,soci\’et\’e math\’ematique de france (1982).

[EN] J. A. Eagon and D. G. Northcott, Ideals defined by matrices and a certaincomplex associated with them, Proc. Roy. Soc. London Ser A 269 (1962), 188-204.

[Gr] J. A. Green, “Polynomial representations of $GL_{n}$ , Lect. Notes Math. 830,Springer Verlag (1980).

[Hasl] M. Hashimoto, Determinantal ideals without minimal free resolutions, NagoyaMath. J. 118 (1990), 203-216.

[Has2] M. Hashimoto, Resolutions of determinantal ideals: t-minors of $(t+2)\cross n$

matrices, J. Alg. 142 (1991), 456-491.

[Has3] M. Hashimoto, Second syzygy of determinantal ideals generated by minors ofgeneric symmetric matrices, in preparation.

[Has4] M. Hashimoto, Relations on Pfaffians: number of generators, preprint.

[HK] M. Hashimoto and K. Kurano, Resolutions of determinantal ideals: n-minors of$(n+2)$-square matrices, Adv. in Math. 94 (1992), 1-66.

[Hib] T. Hibi, Distributive Lattices, Affine Semigroup Rings and Algebras withStraightening Laws, in “Commutative Algebra and Combinatorics,” AdvancedStudies in Pure Mathematics 11, Kinokuniya (1987), 93-109.

[HE] M. Hochster and J. A. Eagon, Cohen-Macaulay rings, invariant theory, and thegeneric perfections of determinantal loci, Amer. J. Math. 93 (1971), 1020-1059.

[Ito] S. Ito, On normality of ASL domains, Hiroshima Math. J. 16 (1986), 607-612.

[Kur] K. Kurano, The first syzygies of determinantal ideals, J. Alg. 124 (1989), 414-436.

[Las] A. Lascoux, Syzygies des varieti\’es d\’eterminantales, Adv. in Math. 30 (1978),202-237.

[CRT] H. Matsumura, “Commutative ring theory” (paperback edition), Cambridge(1989).

[Nls] H. A. Nielsen, Tensor functors of complexes, Aarhus university preprint series15 (1978).

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[PW1] P. Pragacz and J. Weyman, Complexes associated with trace and evaluation.Anothor approach to Lascoux’s resolution, Adv. in Math. 57 (1985), 163-207.

[PW2] P. Pragacz and J. Weyman, On the construction of resolutions of determinantalideals: A survey, in Lect. Notes Math. 1220, Springer Verlag.

[Rob] P. Roberts, “Homological invariants of modules over commutative rings,” LesPresses de l’Universit\’e de Montreal, Montreal (1980).

Department of MathematicsSchool of ScienceNagoya UniversityChikusa-ku, Nagoya 464 JAPAN